Let \({\mathcal {V}}_n({\mathbb {U}},U_{a,b})\) , \(n=0,1,\ldots \) , be the space of orthogonal polynomials of degree \(n\) 
with respect to the inner product \({\langle }\cdot ,\cdot {\rangle }_{{\mathbb {U}}}\)  on the parabolic domain. Then
\(\dim {\mathcal {V}}_n({\mathbb {U}},U_{a,b}) = n+1.\) 
The orthogonal basis for \({\mathcal {V}}_n({\mathbb {U}},U_{a,b})\)  given in [1]} consists of polynomials
\(P_{k,n}^{a,b}(x_1,x_2) = P_{n-k}^{(b+k,a)}(1-2x_2) x_2^{\frac{k}{2}} P_k^{(b-\frac{1}{2},b-\frac{1}{2})}\left(\frac{x_1}{\sqrt{x_2}}\right), \quad 0 \le k \le n.\) 
